Part IB. Quantum Mechanics. Year


 Nancy Rogers
 2 years ago
 Views:
Transcription
1 Part IB Year
2 Paper 4, Section I 6B (a) Define the quantum orbital angular momentum operator ˆL = (ˆL 1, ˆL 2, ˆL 3 ) in three dimensions, in terms of the position and momentum operators. (b) Show that [ˆL 1, ˆL 2 ] = i ˆL 3. [You may assume that the position and momentum operators satisfy the canonical commutation relations.] (c) Let ˆL 2 = ˆL ˆL ˆL 2 3. Show that ˆL 1 commutes with ˆL 2. [In this part of the question you may additionally assume without proof the permuted relations [ˆL 2, ˆL 3 ] = i ˆL 1 and [ˆL 3, ˆL 1 ] = i ˆL 2.] [Hint: It may be useful to consider the expression [Â, ˆB] ˆB + ˆB [Â, ˆB] for suitable operators Â and ˆB.] (d) Suppose that ψ 1 (x, y, z) and ψ 2 (x, y, z) are normalised eigenstates of ˆL 1 with eigenvalues and respectively. Consider the wavefunction ψ = ψ 1 cos ωt + 2 ψ 2 sin ωt, with ω being a positive constant. Find the earliest time t 0 > 0 such that the expectation value of ˆL 1 in ψ is zero. Paper 3, Section I 8B (a) Consider a quantum particle moving in one space dimension, in a timeindependent real potential V (x). For a wavefunction ψ(x, t), define the probability density ρ(x, t) and probability current j(x, t) and show that ρ t + j x = 0. (b) Suppose now that V (x) = 0 and ψ(x, t) = (e ikx + Re ikx )e iet/, where E = 2 k 2 /(2m), k and m are real positive constants, and R is a complex constant. Compute the probability current for this wavefunction. Interpret the terms in ψ and comment on how this relates to the computed expression for the probability current. Part IB, 2016 List of Questions [TURN OVER
3 Paper 1, Section II 15B (a) A particle of mass m in one space dimension is confined to move in a potential V (x) given by { 0 for 0 < x < a, V (x) = for x < 0 or x > a. The normalised initial wavefunction of the particle at time t = 0 is ψ 0 (x) = 4 ( sin 3 πx ). 5a a (i) Find the expectation value of the energy at time t = 0. (ii) Find the wavefunction of the particle at time t = 1. [Hint: It may be useful to recall the identity sin 3θ = 3 sin θ 4 sin 3 θ.] (b) The right hand wall of the potential is lowered to a finite constant value U 0 > 0 giving the new potential: 0 for 0 < x < a, U(x) = for x < 0, U 0 for x > a. This potential is set up in the laboratory but the value of U 0 is unknown. The stationary states of the potential are investigated and it is found that there exists exactly one bound state. Show that the value of U 0 must satisfy π 2 2 8ma 2 < U 0 < 9π2 2 8ma 2. Part IB, 2016 List of Questions
4 Paper 3, Section II 16B The spherically symmetric bound state wavefunctions ψ(r) for the Coulomb potential V = e 2 /(4πǫ 0 r) are normalisable solutions of the equation d 2 ψ dr dψ r dr + 2λ r ψ = 2mE 2 ψ. Here λ = (me 2 )/(4πǫ 0 2 ) and E < 0 is the energy of the state. (a) By writing the wavefunction as ψ(r) = f(r) exp( Kr), for a suitable constant K that you should determine, show that there are normalisable wavefunctions ψ(r) only for energies of the form me 4 E = 32π 2 ǫ N 2, with N being a positive integer. (b) The energies in (a) reproduce the predictions of the Bohr model of the hydrogen atom. How do the wavefunctions above compare to the assumptions in the Bohr model? Part IB, 2016 List of Questions [TURN OVER
5 2016 Paper 2, Section II 17B The one dimensional quantum harmonic oscillator has Hamiltonian 44 Ĥ = 1 2m ˆp mω2ˆx 2, where m and ω are real positive constants and ˆx and ˆp are the standard position and momentum operators satisfying the commutation relation [ˆx, ˆp] = i. Consider the operators Â = ˆp imωˆx and ˆB = ˆp + imωˆx. (a) Show that ˆBÂ = 2m ( Ĥ 1 2 ω ) and Â ˆB = 2m (Ĥ + 12 ) ω. (b) Suppose that φ is an eigenfunction of Ĥ with eigenvalue E. Show that then also an eigenfunction of Ĥ and that its corresponding eigenvalue is E ω. (c) Show that for any normalisable wavefunctions χ and ψ, Âφ is χ (Âψ) dx = ( ˆBχ) ψ dx. [You may assume that the operators ˆx and ˆp are Hermitian.] (d) With φ as in (b), obtain an expression for the norm of Âφ in terms of E and the norm of φ. [The squared norm of any wavefunction ψ is ψ 2 dx.] (e) Show that all eigenvalues of Ĥ are nonnegative. (f) Using the above results, deduce that each eigenvalue E of Ĥ must be of the form E = (n ) ω for some nonnegative integer n. Part IB, 2016 List of Questions
6 Paper 4, Section I 6D The radial wavefunction R(r) for an electron in a hydrogen atom satisfies the equation 2 d 2mr 2 dr (r 2 ddr ) R(r) + 2 l(l + 1)R(r) e2 2mr2 R(r) = E R(r) 4πǫ 0 r ( ) Briefly explain the origin of each term in this equation. The wavefunctions for the ground state and the first radially excited state, both with l = 0, can be written as R 1 (r) = N 1 e αr R 2 (r) = N 2 ( rα) e 1 2 αr where N 1 and N 2 are normalisation constants. Verify that R 1 (r) is a solution of ( ), determining α and finding the corresponding energy eigenvalue E 1. Assuming that R 2 (r) is a solution of ( ), compare coefficients of the dominant terms when r is large to determine the corresponding energy eigenvalue E 2. [You do not need to find N 1 or N 2, nor show that R 2 is a solution of ( ).] A hydrogen atom makes a transition from the first radially excited state to the ground state, emitting a photon. What is the angular frequency of the emitted photon? Paper 3, Section I 8D A quantummechanical system has normalised energy eigenstates χ 1 and χ 2 with nondegenerate energies E 1 and E 2 respectively. The observable A has normalised eigenstates, φ 1 = C(χ 1 + 2χ 2 ), eigenvalue = a 1, φ 2 = C(2χ 1 χ 2 ), eigenvalue = a 2, where C is a positive real constant. Determine C. Initially, at time t = 0, the state of the system is φ 1. Write down an expression for ψ(t), the state of the system with t 0. What is the probability that a measurement of energy at time t will yield E 2? For the same initial state, determine the probability that a measurement of A at time t > 0 will yield a 1 and the probability that it will yield a 2. Part IB, 2015 List of Questions
7 Paper 1, Section II 15D Write down expressions for the probability density ρ(x, t) and the probability current j(x, t) for a particle in one dimension with wavefunction Ψ(x, t). If Ψ(x, t) obeys the timedependent Schrödinger equation with a real potential, show that j x + ρ t = 0. Consider a stationary state, Ψ(x, t) = ψ(x)e iet/, with { e ik 1 ψ(x) x + Re ik 1x x T e ik 2x x +, where E, k 1, k 2 are real. x. Consider a real potential, Evaluate j(x, t) for this state in the regimes x + and V (x) = αδ(x) + U(x), U(x) = { 0 x < 0 V 0 x > 0, where δ(x) is the Dirac delta function, V 0 > 0 and α > 0. Assuming that ψ(x) is continuous at x = 0, derive an expression for [ lim ψ (ǫ) ψ ( ǫ) ]. ǫ 0 Hence calculate the reflection and transmission probabilities for a particle incident from x = with energy E > V 0. Part IB, 2015 List of Questions [TURN OVER
8 Paper 3, Section II 16D Define the angular momentum operators ˆL i for a particle in three dimensions in terms of the position and momentum operators ˆx i and ˆp i = i Write down an expression for [ˆL i, ˆL j ] and use this to show that [ˆL 2, ˆL i ] = 0 where ˆL 2 = ˆL 2 x + ˆL 2 y + ˆL 2 z. What is the significance of these two commutation relations? Let ψ(x, y, z) be both an eigenstate of ˆL z with eigenvalue zero and an eigenstate of ˆL 2 with eigenvalue 2 l(l + 1). Show that (ˆL x + iˆl y )ψ is also an eigenstate of both ˆL z and ˆL 2 and determine the corresponding eigenvalues. Find real constants A and B such that x i. φ(x, y, z) = ( Az 2 + By 2 r 2) e r, r 2 = x 2 + y 2 + z 2, is an eigenfunction of ˆL z with eigenvalue zero and an eigenfunction of ˆL 2 with an eigenvalue which you should determine. [Hint: You might like to show that ˆL i f(r) = 0.] Paper 2, Section II 17D A quantummechanical harmonic oscillator has Hamiltonian where k is a positive real constant. operators. The eigenfunctions of ( ) can be written as Ĥ = ˆp k2ˆx 2. ( ), ψ n (x) = h n (x ) k/ Show that ˆx = x and ˆp = i x exp ) ( kx2, 2 are Hermitian where h n is a polynomial of degree n with even (odd) parity for even (odd) n and n = 0, 1, 2,.... Show that ˆx = ˆp = 0 for all of the states ψ n. State the Heisenberg uncertainty principle and verify it for the state ψ 0 by computing ( x) and ( p). [Hint: You should properly normalise the state.] The oscillator is in its ground state ψ 0 when the potential is suddenly changed so that k 4k. If the wavefunction is expanded in terms of the energy eigenfunctions of the new Hamiltonian, φ n, what can be said about the coefficient of φ n for odd n? What is the probability that the particle is in the new ground state just after the change? [Hint: You may assume that if I n = e ax2 x n dx then I 0 = π a and I 2 = 1 2a π a.] Part IB, 2015 List of Questions
9 Paper 4, Section I 6A For some quantum mechanical observable Q, prove that its uncertainty ( Q) satisfies ( Q) 2 = Q 2 Q 2. A quantum mechanical harmonic oscillator has Hamiltonian H = p2 2m + mω2 x 2, 2 where m > 0. Show that (in a stationary state of energy E) E ( p)2 2m + mω2 ( x) 2. 2 Write down the Heisenberg uncertainty relation. Then, use it to show that for our stationary state. E 1 2 ω Paper 3, Section I 8A The wavefunction of a normalised Gaussian wavepacket for a particle of mass m in one dimension with potential V (x) = 0 is given by ψ(x, t) = B ( x 2 ) A(t) A(t) exp, 2 where A(0) = 1. Given that ψ(x, t) is a solution of the timedependent Schrödinger equation, find the complexvalued function A(t) and the real constant B. [You may assume that e λx2 dx = π/ λ.] Part IB, 2014 List of Questions
10 Paper 1, Section II 15A Consider a particle confined in a onedimensional infinite potential well: V (x) = for x a and V (x) = 0 for x < a. The normalised stationary states are ( ) πn(x + a) α n sin for x < a ψ n (x) = 2a 0 for x a where n = 1, 2,.... (i) Determine the α n and the stationary states energies E n. (ii) A state is prepared within this potential well: ψ(x) x for 0 < x < a, but ψ(x) = 0 for x 0 or x a. Find an explicit expansion of ψ(x) in terms of ψ n (x). (iii) If the energy of the state is then immediately measured, show that the probability that it is greater than 2 π 2 is ma 2 4 n=0 b n π n, where the b n are integers which you should find. (iv) By considering the normalisation condition for ψ(x) in terms of the expansion in ψ n (x), show that π 2 3 = p=1 A p 2 + ) B 2 (1 (2p 1) 2 + C( 1)p, (2p 1)π where A, B and C are integers which you should find. Part IB, 2014 List of Questions [TURN OVER
11 2014 Paper 3, Section II 16A The Hamiltonian of a twodimensional isotropic harmonic oscillator is given by H = p2 x + p 2 y 2m 40 + mω2 2 (x2 + y 2 ), where x and y denote position operators and p x and p y the corresponding momentum operators. State without proof the commutation relations between the operators x, y, p x, p y. From these commutation relations, write [x 2, p x ] and [x, p 2 x ] in terms of a single operator. Now consider the observable L = xp y yp x. Ehrenfest s theorem states that, for some observable Q with expectation value Q, d Q dt = 1 [Q, H] + Q i t. Use it to show that the expectation value of L is constant with time. Given two states ψ 1 = αx exp ( β(x 2 + y 2 ) ) and ψ 2 = αy exp ( β(x 2 + y 2 ) ), where α and β are constants, find a normalised linear combination of ψ 1 and ψ 2 that is an eigenstate of L, and the corresponding L eigenvalue. [You may assume that α correctly normalises both ψ 1 and ψ 2.] If a quantum state is prepared in the linear combination you have found at time t = 0, what is the expectation value of L at a later time t? Part IB, 2014 List of Questions
12 Paper 2, Section II 17A For an electron of mass m in a hydrogen atom, the timeindependent Schrödinger equation may be written as 2 2mr 2 r ( r 2 ψ ) + 1 r 2mr 2 L2 ψ Consider normalised energy eigenstates of the form ψ lm (r, θ, φ) = R(r)Y lm (θ, φ) where Y lm are orbital angular momentum eigenstates: π θ=0 e2 4πǫ 0 r ψ = Eψ. L 2 Y lm = 2 l(l + 1)Y lm, L 3 Y lm = my lm, where l = 1, 2,... and m = 0, ±1, ±2,... ± l. The Y lm functions are normalised with 2π φ=0 Y lm 2 sin θ dθ dφ = 1. (i) Write down the resulting equation satisfied by R(r), for fixed l. Show that it has solutions of the form ( R(r) = Ar l exp r ), a(l + 1) where a is a constant which you should determine. Show that e2 E = Dπǫ 0 a, where D is an integer which you should find (in terms of l). Also, show that A 2 = 2 2l+3 a F G!(l + 1) H, where F, G and H are integers that you should find in terms of l. (ii) Given the radius of the proton r p a, show that the probability of the electron being found within the proton is approximately 2 2l+3 finding the integer C in terms of l. [You may assume that 0 t l e t dt = l!.] C ( rp ) 2l+3 [ ( rp )] 1 + O, a a Part IB, 2014 List of Questions [TURN OVER
13 Paper 4, Section I 6B The components of the threedimensional angular momentum operator ˆL are defined as follows: ( ˆL x = i y z z ) ( ˆLy = i z y x x ) ˆLz = i ( x z y y ). x Given that the wavefunction ψ = (f(x) + iy)z is an eigenfunction of ˆL z, find all possible values of f(x) and the corresponding eigenvalues of ψ. Letting f(x) = x, show that ψ is an eigenfunction of ˆL 2 and calculate the corresponding eigenvalue. Paper 3, Section I 8B If α, β and γ are linear operators, establish the identity [αβ, γ] = α[β, γ] + [α, γ]β. In what follows, the operators x and p are Hermitian and represent position and momentum of a quantum mechanical particle in onedimension. Show that and [x n, p] = i nx n 1 [x, p m ] = i mp m 1 where m, n Z +. Assuming [x n, p m ] 0, show that the operators x n and p m are Hermitian but their product is not. Determine whether x n p m + p m x n is Hermitian. Part IB, 2013 List of Questions
14 Paper 1, Section II 15B A particle with momentum ˆp moves in a onedimensional real potential with Hamiltonian given by Ĥ = 1 (ˆp + isa)(ˆp isa), 2m < x < where A is a real function and s R +. Obtain the potential energy of the system. Find χ(x) such that (ˆp isa)χ(x) = 0. Now, putting A = x n, for n Z +, show that χ(x) can be normalised only if n is odd. Letting n = 1, use the inequality to show that assuming that both ˆp and ˆx vanish. ψ (x)ĥψ(x)dx 0 x p 2 Paper 3, Section II 16B Obtain, with the aid of the timedependent Schrödinger equation, the conservation equation ρ(x, t) + j(x, t) = 0 t where ρ(x, t) is the probability density and j(x, t) is the probability current. What have you assumed about the potential energy of the system? Show that if the potential U(x, t) is complex the conservation equation becomes t ρ(x, t) + j(x, t) = 2 ρ(x, t) ImU(x, t). Take the potential to be timeindependent. Show, with the aid of the divergence theorem, that d ρ(x, t) dv = 2 ρ(x, t) ImU(x) dv. dt R 3 R 3 Assuming the wavefunction ψ(x, 0) is normalised to unity, show that if ρ(x, t) is expanded about t = 0 so that ρ(x, t) = ρ 0 (x) + tρ 1 (x) +, then ρ(x, t) dv = 1 + 2t ρ 0 (x) ImU(x) dv +. R 3 R 3 As time increases, how does the quantity on the left of this equation behave if ImU(x) < 0? Part IB, 2013 List of Questions [TURN OVER
15 Paper 2, Section II 17B (i) Consider a particle of mass m confined to a onedimensional potential well of depth U > 0 and potential { U, x < l V (x) = 0, x > l. If the particle has energy E where U E < 0, show that for even states α tan αl = β where α = [ 2m 2 (U + E)] 1/2 and β = [ 2m 2 E] 1/2. (ii) A particle of mass m that is incident from the left scatters off a onedimensional potential given by V (x) = kδ(x) where δ(x) is the Dirac delta. If the particle has energy E > 0 and k > 0, obtain the reflection and transmission coefficients R and T, respectively. Confirm that R + T = 1. For the case k < 0 and E < 0 show that the energy of the only even parity bound state of the system is given by E = mk Use part (i) to verify this result by taking the limit U, l 0 with Ul fixed. Part IB, 2013 List of Questions
16 2012 Paper 4, Section I 6C In terms of quantum states, what is meant by energy degeneracy? 39 A particle of mass m is confined within the box 0 < x < a, 0 < y < a and 0 < z < c. The potential vanishes inside the box and is infinite outside. Find the allowed energies by considering a stationary state wavefunction of the form χ(x, y, z) = X(x) Y (y) Z(z). Write down the normalised ground state wavefunction. Assuming that c < a < 2c, give the energies of the first three excited states. Paper 3, Section I 8C A onedimensional quantum mechanical particle has normalised bound state energy eigenfunctions χ n (x) and corresponding nondegenerate energy eigenvalues E n. At t = 0 the normalised wavefunction ψ(x, t) is given by ψ(x, 0) = 5 6 eik 1 1 χ 1 (x) + 6 eik 2 χ 2 (x) where k 1 and k 2 are real constants. Write down the expression for ψ(x, t) at a later time t and give the probability that a measurement of the particle s energy will yield a value of E 2. Show that the expectation value of x at time t is given by x = 5 6 x [ ] 6 x Re x 12 e i(k 2 k 1 ) i(e 2 E 1 )t/ where x ij = χ i (x) x χ j(x) dx. Part IB, 2012 List of Questions [TURN OVER
17 Paper 1, Section II 15C Show that if the energy levels are discrete, the general solution of the Schrödinger equation i ψ t = 2 2m 2 ψ + V (x)ψ is a linear superposition of stationary states ψ(x, t) = a n χ n (x) exp( ie n t/ ), n=1 where χ n (x) is a solution of the timeindependent Schrödinger equation and a n are complex coefficients. Can this general solution be considered to be a stationary state? Justify your answer. A linear operator Ô acts on the orthonormal energy eigenfunctions χ n as follows: Ôχ 1 = χ 1 + χ 2 Ôχ 2 = χ 1 + χ 2 Ôχ n = 0, n 3. Obtain the eigenvalues of Ô. Hence, find the normalised eigenfunctions of Ô. In an experiment a measurement is made of Ô at t = 0 yielding an eigenvalue of 2. What is the probability that a measurement at some later time t will yield an eigenvalue of 2? Paper 3, Section II 16C State the condition for a linear operator Ô to be Hermitian. Given the position and momentum operators ˆx i and ˆp i = i x i, define the angular momentum operators ˆL i. Establish the commutation relations [ˆL i, ˆL j ] = i ǫ ijk ˆLk and use these relations to show that ˆL 3 is Hermitian assuming ˆL 1 and ˆL 2 are. Consider a wavefunction of the form χ(x) = x 3 (x 1 + kx 2 )e r where r = x and k is some constant. Show that χ(x) is an eigenstate of the total angular momentum operator ˆL 2 for all k, and calculate the corresponding eigenvalue. For what values of k is χ(x) an eigenstate of ˆL 3? What are the corresponding eigenvalues? Part IB, 2012 List of Questions
18 2012 Paper 2, Section II 17C 41 Consider a quantum mechanical particle in a onedimensional potential V (x), for which V (x) = V ( x). Prove that when the energy eigenvalue E is nondegenerate, the energy eigenfunction χ(x) has definite parity. Now assume the particle is in the double potential well U, 0 x l 1 V (x) = 0, l 1 < x l 2, l 2 < x, where 0 < l 1 < l 2 and 0 < E < U (U being large and positive). Obtain general expressions for the even parity energy eigenfunctions χ + (x) in terms of trigonometric and hyperbolic functions. Show that where k 2 = 2mE 2 and κ 2 2m(U E) = 2. tan[k(l 2 l 1 )] = k κ coth(κl 1), Part IB, 2012 List of Questions [TURN OVER
19 Paper 3, Section I 8C A particle of mass m and energy E, incident from x =, scatters off a delta function potential at x = 0. The time independent Schrödinger equation is 2 d 2 ψ + Uδ(x)ψ = Eψ 2m dx2 where U is a positive constant. Find the reflection and transmission probabilities. Paper 4, Section I 6C Consider the 3dimensional oscillator with Hamiltonian H = 2 2m 2 + mω2 2 (x2 + y 2 + 4z 2 ). Find the ground state energy and the spacing between energy levels. Find the degeneracies of the lowest three energy levels. [You may assume that the energy levels of the 1dimensional harmonic oscillator with Hamiltonian H 0 = 2 d 2 2m dx 2 + mω2 2 x2 are (n ) ω, n = 0, 1, 2,....] Part IB, 2011 List of Questions [TURN OVER
20 Paper 1, Section II 15C For a quantum mechanical particle moving freely on a circle of length 2π, the wavefunction ψ(t, x) satisfies the Schrödinger equation i ψ t = 2 2 ψ 2m x 2 on the interval 0 x 2π, and also the periodicity conditions ψ(t, 2π) = ψ(t, 0), and ψ ψ (t, 2π) = (t, 0). Find the allowed energy levels of the particle, and their degeneracies. x x The current is defined as j = i ( ψ ψ ψ ψ ) 2m x x where ψ is a normalized state. Write down the general normalized state of the particle when it has energy 2 2 /m, and show that in any such state the current j is independent of x and t. Find a state with this energy for which the current has its maximum positive value, and find a state with this energy for which the current vanishes. Paper 2, Section II 17C The quantum mechanical angular momentum operators are L i = i ǫ ijk x j Show that each of these is hermitian. x k (i = 1, 2, 3). The total angular momentum operator is defined as L 2 = L L2 2 + L2 3. Show that L 2 L 2 3 in any state, and show that the only states where L2 = L 2 3 are those with no angular dependence. Verify that the eigenvalues of the operators L 2 and L 2 3 (whose values you may quote without proof) are consistent with these results. Part IB, 2011 List of Questions
21 Paper 3, Section II 16C For an electron in a hydrogen atom, the stationary state wavefunctions are of the form ψ(r, θ, φ) = R(r)Y lm (θ, φ), where in suitable units R obeys the radial equation d 2 R dr ( dr l(l + 1) r dr r 2 R + 2 E + 1 ) R = 0. r Explain briefly how the terms in this equation arise. This radial equation has bound state solutions of energy E = E n, where E n = 1 (n = 1, 2, 3,... ). Show that when l = n 1, there is a solution of the 2n 2 form R(r) = r α e r/n, and determine α. Find the expectation value r in this state. What is the total degeneracy of the energy level with energy E n? Part IB, 2011 List of Questions [TURN OVER
22 Paper 3, Section I 8D Write down the commutation relations between the components of position x and momentum p for a particle in three dimensions. A particle of mass m executes simple harmonic motion with Hamiltonian H = 1 2m p2 + mω2 2 x2, and the orbital angular momentum operator is defined by L = x p. Show that the components of L are observables commuting with H. Explain briefly why the components of L are not simultaneous observables. What are the implications for the labelling of states of the threedimensional harmonic oscillator? Paper 4, Section I 6D Determine the possible values of the energy of a particle free to move inside a cube of side a, confined there by a potential which is infinite outside and zero inside. What is the degeneracy of the lowestbutone energy level? Part IB, 2010 List of Questions [TURN OVER
23 2010 Paper 1, Section II 15D A particle of unit mass moves in one dimension in a potential 38 V = 1 2 ω2 x 2. Show that the stationary solutions can be written in the form ψ n (x) = f n (x) exp( αx 2 ). You should give the value of α and derive any restrictions on f n (x). Hence determine the possible energy eigenvalues E n. The particle has a wave function ψ(x, t) which is even in x at t = 0. Write down the general form for ψ(x, 0), using the fact that f n (x) is an even function of x only if n is even. Hence write down ψ(x, t) and show that its probability density is periodic in time with period π/ω. Paper 2, Section II 17D A particle of mass m moves in a onedimensional potential defined by for x < 0, V (x) = 0 for 0 x a, V 0 for a < x, where a and V 0 are positive constants. Defining c = [2m(V 0 E)] 1/2 / and k = (2mE) 1/2 /, show that for any allowed positive value E of the energy with E < V 0 then c + k cot ka = 0. Find the minimum value of V 0 for this equation to have a solution. Find the normalized wave function for the particle. Write down an expression for the expectation value of x in terms of two integrals, which you need not evaluate. Given that x = 1 (ka tan ka), 2k discuss briefly the possibility of x being greater than a. [Hint: consider the graph of ka cot ka against ka.] Part IB, 2010 List of Questions
24 Paper 3, Section II 16D A π (a particle of the same charge as the electron but 270 times more massive) is bound in the Coulomb potential of a proton. Assuming that the wave function has the form ce r/a, where c and a are constants, determine the normalized wave function of the lowest energy state of the π, assuming it to be an Swave (i.e. the state with l = 0). (You should treat the proton as fixed in space.) Calculate the probability of finding the π inside a sphere of radius R in terms of the ratio µ = R/a, and show that this probability is given by 4µ 3 /3 + O(µ 4 ) if µ is very small. Would the result be larger or smaller if the π were in a P wave (l = 1) state? Justify your answer very briefly. [Hint: in spherical polar coordinates, 2 ψ(r) = 1 r 2 r 2 (rψ) + 1 r 2 sin θ ( sin θ ψ ) + θ θ 1 2 ] ψ r 2 sin 2 θ φ 2. Part IB, 2010 List of Questions [TURN OVER
25 2009 Paper 3, Section I 7B 42 The motion of a particle in one dimension is described by the timeindependent hermitian Hamiltonian operator H whose normalized eigenstates ψ n (x), n = 0, 1, 2,..., satisfy the Schrödinger equation Hψ n = E n ψ n, with E 0 < E 1 < E 2 < < E n <. Show that ψ mψ n dx = δ mn. The particle is in a state represented by the wavefunction Ψ(x, t) which, at time t = 0, is given by ( ) 1 n+1 Ψ(x, 0) = ψ n (x). 2 Write down an expression for Ψ(x, t) and show that it is normalized to unity. n=0 Derive an expression for the expectation value of the energy for this state and show that it is independent of time. Calculate the probability that the particle has energy E m for a given integer m 0, and show that this also is timeindependent. Paper 4, Section I 6B The wavefunction of a Gaussian wavepacket for a particle of mass m moving in one dimension is ψ(x, t) = 1 ( 1 π 1/4 1 + i t/m exp x 2 ). 2(1 + i t/m) Show that ψ(x, t) satisfies the appropriate timedependent Schrödinger equation. Show that ψ(x, t) is normalized to unity and calculate the uncertainty in measurement of the particle position, x = x 2 x 2. Is ψ(x, t) a stationary state? Give a reason for your answer. [ You may assume that e λx2 dx = ] π λ. Part IB, 2009 List of Questions
26 2009 Paper 1, Section II 15B 43 A particle of mass m moves in one dimension in a potential V (x) which satisfies V (x) = V ( x). Show that the eigenstates of the Hamiltonian H can be chosen so that they are also eigenstates of the parity operator P. For eigenstates with odd parity ψ odd (x), show that ψ odd (0) = 0. A potential V (x) is given by V (x) = { κδ(x) x < a x > a. State the boundary conditions satisfied by ψ(x) at x = a, and show also that 2 [ dψ 2m lim ǫ 0 dx dψ ] ǫ dx = κψ(0). ǫ Let the energy eigenstates of even parity be given by A cos λx + B sin λx a < x < 0 ψ even (x) = A cos λx B sin λx 0 < x < a 0 otherwise. Verify that ψ even (x) satisfies P ψ even (x) = ψ even (x). satisfy By demanding that ψ even (x) satisfy the relevant boundary conditions show that tan λa = 2 λ m κ. For κ > 0 show that the energy eigenvalues E even n Show also that η n = E even n 1 2m [ ] (2n + 1) π 2 > 0. 2a lim η n = 0, n and give a physical explanation of this result., n = 0, 1, 2,..., with E even n < E even n+1, Show that the energy eigenstates with odd parity and their energy eigenvalues do not depend on κ. Part IB, 2009 List of Questions [TURN OVER
27 2009 Paper 2, Section II 16B 44 Write down the expressions for the probability density ρ and the associated current density j for a particle with wavefunction ψ(x, t) moving in one dimension. If ψ(x, t) obeys the timedependent Schrödinger equation show that ρ and j satisfy j x + ρ t = 0. Give an interpretation of ψ(x, t) in the case that ψ(x, t) = (e ikx + Re ikx )e iet/, and show that E = 2 k 2 ρ and 2m t = 0. A particle of mass m and energy E > 0 moving in one dimension is incident from the left on a potential V (x) given by { V0 0 < x < a V (x) = 0 x < 0, x > a, where V 0 is a positive constant. What conditions must be imposed on the wavefunction at x = 0 and x = a? Show that when 3E = V 0 the probability of transmission is [ a ] 1 8mE 16 sin2. For what values of a does this agree with the classical result? Part IB, 2009 List of Questions
28 Paper 3, Section II 16B If A, B, and C are operators establish the identity [AB, C] = A[B, C] + [A, C]B. A particle moves in a twodimensional harmonic oscillator potential with Hamiltonian H = 1 2 (p2 x + p 2 y) (x2 + y 2 ). The angular momentum operator is defined by L = xp y yp x. Show that L is hermitian and hence that its eigenvalues are real. Establish the commutation relation [L, H] = 0. Why does this ensure that eigenstates of H can also be chosen to be eigenstates of L? Let φ 0 (x, y) = e (x2 +y 2 )/2, and show that φ 0, φ x = xφ 0 and φ y = yφ 0 are all eigenstates of H, and find their respective eigenvalues. Show that Lφ 0 = 0, Lφ x = i φ y, Lφ y = i φ x, and hence, by taking suitable linear combinations of φ x and φ y, find two states, ψ 1 and ψ 2, satisfying Lψ j = λ j ψ j, Hψ j = E j ψ j j = 1, 2. Show that ψ 1 and ψ 2 are orthogonal, and find λ 1, λ 2, E 1 and E 2. The particle has charge e, and an electric field of strength E is applied in the x direction so that the Hamiltonian is now H, where H = H eex. Show that [L, H ] = i eey. Why does this mean that L and H cannot have simultaneous eigenstates? By making the change of coordinates x = x ee, y = y, show that ψ 1 (x, y ) and ψ 2 (x, y ) are eigenstates of H and write down the corresponding energy eigenvalues. Find a modified angular momentum operator L for which ψ 1 (x, y ) and ψ 2 (x, y ) are also eigenstates. Part IB, 2009 List of Questions [TURN OVER
29 /II/15A The radial wavefunction g(r) for the hydrogen atom satisfies the equation 2 2mr 2 ( d r 2 dg(r) ) e2 g(r) dr dr 4πɛ 0 r l(l + 1) + 2 g(r) = Eg(r). ( ) 2mr2 With reference to the general form for the timeindependent Schrödinger equation, explain the origin of each term. What are the allowed values of l? The lowestenergy boundstate solution of ( ), for given l, has the form r α e βr. Find α and β and the corresponding energy E in terms of l. A hydrogen atom makes a transition between two such states corresponding to l+1 and l. What is the frequency of the emitted photon? 2/II/16A Give the physical interpretation of the expression A ψ = ψ(x) Âψ(x)dx for an observable A, where Â is a Hermitian operator and ψ is normalised. By considering the norm of the state (A + iλb)ψ for two observables A and B, and real values of λ, show that A 2 ψ B 2 ψ 1 4 [A, B] ψ 2. Deduce the uncertainty relation where A is the uncertainty of A. A B 1 2 [A, B] ψ, A particle of mass m moves in one dimension under the influence of potential 1 2 mω2 x 2. By considering the commutator [x, p], show that the expectation value of the Hamiltonian satisfies H ψ 1 2 ω. Part IB 2008
30 /I/7A Write down a formula for the orbital angular momentum operator ˆL. Show that its components satisfy [L i, L j ] = i ɛ ijk L k. If L 3 ψ = 0, show that (L 1 ± il 2 )ψ are also eigenvectors of L 3, and find their eigenvalues. 3/II/16A What is the probability current for a particle of mass m, wavefunction ψ, moving in one dimension? A particle of energy E is incident from x < 0 on a barrier given by 0 x 0 V (x) = V 1 0 < x < a x a V 0 where V 1 > V 0 > 0. What are the conditions satisfied by ψ at x = 0 and x = a? Write down the form taken by the wavefunction in the regions x 0 and x a distinguishing between the cases E > V 0 and E < V 0. For both cases, use your expressions for ψ to calculate the probability currents in these two regions. Define the reflection and transmission coefficients, R and T. Using current conservation, show that the expressions you have derived satisfy R + T = 1. Show that T = 0 if 0 < E < V 0. 4/I/6A What is meant by a stationary state? What form does the wavefunction take in such a state? A particle has wavefunction ψ(x, t), such that ψ(x, 0) = 1 2 (χ 1(x) + χ 2 (x)), where χ 1 and χ 2 are normalised eigenstates of the Hamiltonian with energies E 1 and E 2. Write down ψ(x, t) at time t. Show that the expectation value of A at time t is A ψ = 1 2 ( ) ) χ 1Âχ 1 + χ 2Âχ 2 dx + Re (e i(e1 E2)t/ χ 1Âχ 2 dx. Part IB 2008
31 /II/15B The relative motion of a neutron and proton is described by the Schrödinger equation for a single particle of mass m under the influence of the central potential V (r) = { U r < a 0 r > a, where U and a are positive constants. Solve this equation for a spherically symmetric state of the deuteron, which is a bound state of a proton and neutron, giving the condition on U for this state to exist. [If ψ is spherically symmetric then 2 ψ = 1 r d 2 dr 2 (rψ).] 2/II/16B Write down the angular momentum operators L 1, L 2, L 3 in terms of the position and momentum operators, x and p, and the commutation relations satisfied by x and p. Verify the commutation relations [L i, L j ] = i ɛ ijk L k. Further, show that [L i, p j ] = i ɛ ijk p k. A wavefunction Ψ 0 (r) is spherically symmetric. Verify that LΨ 0 (r) = 0. Consider the vector function Φ = Ψ 0 (r). Show that Φ 3 and Φ 1 ± iφ 2 are eigenfunctions of L 3 with eigenvalues 0, ± respectively. Part IB 2007
32 /I/7B The quantum mechanical harmonic oscillator has Hamiltonian H = 1 2m p mω2 x 2, and is in a stationary state of energy < H >= E. Show that E 1 2m ( p) mω2 ( x) 2, where ( p) 2 = p 2 p 2 and ( x) 2 = x 2 x 2. Use the Heisenberg Uncertainty Principle to show that E 1 2 ω. 3/II/16B A quantum system has a complete set of orthonormal eigenstates, ψ n (x), with nondegenerate energy eigenvalues, E n, where n = 1, 2, Write down the wavefunction, Ψ(x, t), t 0 in terms of the eigenstates. A linear operator acts on the system such that Aψ 1 = 2ψ 1 ψ 2 Aψ 2 = 2ψ 2 ψ 1 Aψ n = 0, n 3. Find the eigenvalues of A and obtain a complete set of normalised eigenfunctions, φ n, of A in terms of the ψ n. At time t = 0 a measurement is made and it is found that the observable corresponding to A has value 3. After time t, A is measured again. What is the probability that the value is found to be 1? 4/I/6B A particle moving in one space dimension with wavefunction Ψ(x, t) obeys the timedependent Schrödinger equation. Write down the probability density, ρ, and current density, j, in terms of the wavefunction and show that they obey the equation j x + ρ t = 0. The wavefunction is Ψ(x, t) = ( e ikx + R e ikx) e iet/, where E = 2 k 2 /2m and R is a constant, which may be complex. Evaluate j. Part IB 2007
33 /II/15B Let V 1 (x) and V 2 (x) be two real potential functions of one space dimension, and let a be a positive constant. Suppose also that V 1 (x) V 2 (x) 0 for all x and that V 1 (x) = V 2 (x) = 0 for all x such that x a. Consider an incoming beam of particles described by the plane wave exp(ikx), for some k > 0, scattering off one of the potentials V 1 (x) or V 2 (x). Let p i be the probability that a particle in the beam is reflected by the potential V i (x). Is it necessarily the case that p 1 p 2? Justify your answer carefully, either by giving a rigorous proof or by presenting a counterexample with explicit calculations of p 1 and p 2. 2/II/16B The spherically symmetric bound state wavefunctions ψ(r), where r = x, for an electron orbiting in the Coulomb potential V (r) = e 2 /(4πɛ 0 r) of a hydrogen atom nucleus, can be modelled as solutions to the equation d 2 ψ dr dψ r dr + a r ψ(r) b2 ψ(r) = 0 for r 0, where a = e 2 m/(2πɛ 0 2 ), b = 2mE/, and E is the energy of the corresponding state. Show that there are normalisable and continuous wavefunctions ψ(r) satisfying this equation with energies for all integers N 1. me 4 E = 32π 2 ɛ N 2 3/I/7B Define the quantum mechanical operators for the angular momentum ˆL and the total angular momentum ˆL 2 in terms of the operators ˆx and. Calculate the commutators [ˆL i, ˆL j ] and [ˆL 2, ˆL i ]. Part IB 2006
34 /II/16B The expression ψ A denotes the uncertainty of a quantum mechanical observable A in a state with normalised wavefunction ψ. Prove that the Heisenberg uncertainty principle ( ψ x)( ψ p) 2 holds for all normalised wavefunctions ψ(x) of one spatial dimension. [You may quote Schwarz s inequality without proof.] A Gaussian wavepacket evolves so that at time t its wavefunction is ψ(x, t) = (2π) 1 4 ( ) 1 ( i t exp x 2 4(1 + i t) Calculate the uncertainties ψ x and ψ p at each time t, and hence verify explicitly that the uncertainty principle holds at each time t. [ You may quote without proof the results that if Re(a) > 0 then ). and exp ( d dx exp ( x2 a ( x2 a ) ) x 2 exp ( x2 dx = 1 ( π ) 1 2 a 3 a 4 2 (Re(a)) 3 2 )) ( )) d ( ( dx exp x2 π ) 1 2 a dx = a 2 (Re(a)) 3 2 ]. 4/I/6B (a) Define the probability density ρ(x, t) and the probability current J(x, t) for a quantum mechanical wave function ψ(x, t), where the three dimensional vector x defines spatial coordinates. Given that the potential V (x) is real, show that J + ρ t = 0. (b) Write down the standard integral expressions for the expectation value A ψ and the uncertainty ψ A of a quantum mechanical observable A in a state with wavefunction ψ(x). Give an expression for ψ A in terms of A 2 ψ and A ψ, and justify your answer. Part IB 2006
35 /II/15G The wave function of a particle of mass m that moves in a onedimensional potential well satisfies the Schrödinger equation with a potential that is zero in the region a x a and infinite elsewhere, V (x) = 0 for x a, V (x) = for x > a. Determine the complete set of normalised energy eigenfunctions for the particle and show that the energy eigenvalues are E = 2 π 2 n 2 8ma 2, where n is a positive integer. At time t = 0 the wave function is ψ(x) = 1 5a cos ( πx ) + 2 sin 2a 5a ( πx ), a in the region a x a, and zero otherwise. Determine the possible results for a measurement of the energy of the system and the relative probabilities of obtaining these energies. In an experiment the system is measured to be in its lowest possible energy eigenstate. The width of the well is then doubled while the wave function is unaltered. Calculate the probability that a later measurement will find the particle to be in the lowest energy state of the new potential well. Part IB 2005
36 /II/16G A particle of mass m moving in a onedimensional harmonic oscillator potential satisfies the Schrödinger equation where the Hamiltonian is given by H Ψ(x, t) = i Ψ(x, t), t d 2 H = 2 2m dx m ω2 x 2. The operators a and a are defined by a = 1 ( βx + i ) 2 β p, a = 1 ( βx i ) 2 β p, where β = mω/ and p = i / x is the usual momentum operator. [a, a ] = 1. Show that Express x and p in terms of a and a and, hence or otherwise, show that H can be written in the form H = ( a a + 1 2) ω. Show, for an arbitrary wave function Ψ, that dx Ψ H Ψ 1 2 ω and hence that the energy of any state satisfies the bound E 1 2 ω. Hence, or otherwise, show that the ground state wave function satisfies aψ 0 = 0 and that its energy is given by E 0 = 1 2 ω. By considering H acting on a Ψ 0, (a ) 2 Ψ 0, and so on, show that states of the form (a ) n Ψ 0 (n > 0) are also eigenstates and that their energies are given by E n = ( n + 1 2) ω. Part IB 2005
37 /I/7G The wave function Ψ(x, t) is a solution of the timedependent Schrödinger equation for a particle of mass m in a potential V (x), H Ψ(x, t) = i Ψ(x, t), t where H is the Hamiltonian. Define the expectation value, O, of any operator O. At time t = 0, Ψ(x, t) can be written as a sum of the form Ψ(x, 0) = n a n u n (x), where u n is a complete set of normalized eigenfunctions of the Hamiltonian with energy eigenvalues E n and a n are complex coefficients that satisfy n a na n = 1. Find Ψ(x, t) for t > 0. What is the probability of finding the system in a state with energy E p at time t? Show that the expectation value of the energy is independent of time. 3/II/16G A particle of mass µ moves in two dimensions in an axisymmetric potential. Show that the timeindependent Schrödinger equation can be separated in polar coordinates. Show that the angular part of the wave function has the form e imφ, where φ is the angular coordinate and m is an integer. Suppose that the potential is zero for r < a, where r is the radial coordinate, and infinite otherwise. Show that the radial part of the wave function satisfies d 2 R dρ dr ρ dρ + (1 m2 ρ 2 ) R = 0, where ρ = r ( 2µE/ 2) 1/2. What conditions must R satisfy at r = 0 and R = a? Show that, when m = 0, the equation has the solution R(ρ) = k=0 A k ρ k, where A k = 0 if k is odd and if k is even. A k A k+2 = (k + 2) 2, Deduce the coefficients A 2 and A 4 in terms of A 0. By truncating the series expansion at order ρ 4, estimate the smallest value of ρ at which the R is zero. Hence give an estimate of the ground state energy. [You may use the fact that the Laplace operator is given in polar coordinates by the expression 2 = 2 r r r ] r 2 φ 2. Part IB 2005
38 /I/6G Define the commutator [A, B] of two operators, A and B. In three dimensions angular momentum is defined by a vector operator L with components L x = y p z z p y L y = z p x x p z L z = x p y y p x. Show that [L x, L y ] = i L z and use this, together with permutations, to show that [L 2, L w ] = 0, where w denotes any of the directions x, y, z. At a given time the wave function of a particle is given by ψ = (x + y + z) exp ( ) x 2 + y 2 + z 2. Show that this is an eigenstate of L 2 with eigenvalue equal to 2 2. Part IB 2005
39 /I/8D From the timedependent Schrödinger equation for ψ(x, t), derive the equation ρ t + j x = 0 for ρ(x, t) = ψ (x, t)ψ(x, t) and some suitable j(x, t). Show that ψ(x, t) = e i(kx ωt) is a solution of the timedependent Schrödinger equation with zero potential for suitable ω(k) and calculate ρ and j. What is the interpretation of this solution? 1/II/19D The angular momentum operators are L = (L 1, L 2, L 3 ). commutation relations and show that [L i, L 2 ] = 0. Let Write down their and show that L ± = L 1 ± il 2, L 2 = L L + + L L 3. Verify that Lf(r) = 0, where r 2 = x i x i, for any function f. Show that L 3 (x 1 + ix 2 ) n f(r) = n (x 1 + ix 2 ) n f(r), L + (x 1 + ix 2 ) n f(r) = 0, for any integer n. Show that (x 1 + ix 2 ) n f(r) is an eigenfunction of L 2 and determine its eigenvalue. Why must L (x 1 + ix 2 ) n f(r) be an eigenfunction of L 2? What is its eigenvalue? 2/I/8D A quantum mechanical system is described by vectors ψ = eigenvectors are ψ 0 = ( ) cos θ, ψ sin θ 1 = ( ) sin θ, cos θ with energies E 0, E 1 respectively. The system is in the state ( ) 0 is the probability of finding it in the state at a later time t? 1 ( ) a. The energy b ( ) 1 at time t = 0. What 0 Part IB 2004
40 /II/19D Consider a Hamiltonian of the form H = 1 ( )( ) p + if(x) p if(x), < x <, 2m where f(x) is a real function. Show that this can be written in the form H = p 2 /(2m) + V (x), for some real V (x) to be determined. Show that there is a wave function ψ 0 (x), satisfying a firstorder equation, such that Hψ 0 = 0. If f is a polynomial of degree n, show that n must be odd in order for ψ 0 to be normalisable. By considering dx ψ Hψ show that all energy eigenvalues other than that for ψ 0 must be positive. For f(x) = kx, use these results to find the lowest energy and corresponding wave function for the harmonic oscillator Hamiltonian H oscillator = p2 2m mω2 x 2. 3/I/9D Write down the expressions for the classical energy and angular momentum for an electron in a hydrogen atom. In the Bohr model the angular momentum L is quantised so that L = n, for integer n. Assuming circular orbits, show that the radius of the n th orbit is r n = n 2 a, and determine a. Show that the corresponding energy is then e2 E n =. 8πɛ 0 r n Part IB 2004
41 /II/20D A onedimensional system has the potential { 0 x < 0, V (x) = 2 U 2m 0 < x < L, 0 x > L. For energy E = 2 ɛ/(2m), ɛ < U, the wave function has the form a e ikx + c e ikx x < 0, ψ(x) = e cosh Kx + f sinh Kx 0 < x < L, d e ik(x L) + b e ik(x L) x > L. By considering the relation between incoming and outgoing waves explain why we should expect c 2 + d 2 = a 2 + b 2. Find four linear relations between a, b, c, d, e, f. Eliminate d, e, f and show that c = 1 D [ b + 1 ( λ 1 ) ] sinh KL a, 2 λ where D = cosh KL 1 2( λ + 1 λ) sinh KL and λ = K/(ik). By using the result for c, or otherwise, explain why the solution for d is d = 1 D [ a + 1 ( λ 1 ) ] sinh KL b. 2 λ For b = 0 define the transmission coefficient T and show that, for large L, T 16 ɛ(u ɛ) U 2 e 2 U ɛ L. Part IB 2004
42 /I/9A A particle of mass m is confined inside a onedimensional box of length a. Determine the possible energy eigenvalues. 1/II/18A What is the significance of the expectation value Q = ψ (x) Q ψ(x)dx of an observable Q in the normalized state ψ(x)? Let Q and P be two observables. By considering the norm of (Q + iλp )ψ for real values of λ, show that Q 2 P [Q, P ] 2. The uncertainty Q of Q in the state ψ(x) is defined as Deduce the generalized uncertainty relation, ( Q) 2 = (Q Q ) 2. Q P 1 2 [Q, P ]. A particle of mass m moves in one dimension under the influence of the potential 1 2 mω2 x 2. By considering the commutator [x, p], show that the expectation value of the Hamiltonian satisfies H 1 2 ω. 2/I/9A What is meant by the statement than an operator is hermitian? A particle of mass m moves in the real potential V (x) in one dimension. Show that the Hamiltonian of the system is hermitian. Show that d dt x = 1 m p, d dt p = V (x), where p is the momentum operator and A denotes the expectation value of the operator A. Part IB 2003
43 /II/18A A particle of mass m and energy E moving in one dimension is incident from the left on a potential barrier V (x) given by with V 0 > 0. V (x) = { V0 0 x a 0 otherwise In the limit V 0, a 0 with V 0 a = U held fixed, show that the transmission probability is T = (1 + mu 2 ) 1 2E 2. 3/II/20A The radial wavefunction for the hydrogen atom satisfies the equation 2 2m 1 r 2 d (r 2 ddr ) dr R(r) + 2 l(l + 1)R(r) e2 2mr2 Explain the origin of each term in this equation. R(r) = ER(r). 4πɛ 0 r The wavefunctions for the ground state and first radially excited state, both with l = 0, can be written as R 1 (r) = N 1 exp( αr) R 2 (r) = N 2 (r + b) exp( βr) respectively, where N 1 and N 2 are normalization constants. corresponding energy eigenvalues E 1 and E 2. Determine α, β, b and the A hydrogen atom is in the first radially excited state. It makes the transition to the ground state, emitting a photon. What is the frequency of the emitted photon? Part IB 2003
44 /I/9D Consider a quantum mechanical particle of mass m moving in one dimension, in a potential well, x < 0, V (x) = 0, 0 < x < a, V 0, x > a. Sketch the ground state energy eigenfunction χ(x) and show that its energy is E = 2 k 2 2m, where k satisfies k tan ka =. 2mV 0 k 2 2 [Hint: You may assume that χ(0) = 0. ] 1/II/18D A quantum mechanical particle of mass M moves in one dimension in the presence of a negative delta function potential 2 V = 2M δ(x), where is a parameter with dimensions of length. (a) Write down the timeindependent Schrödinger equation for energy eigenstates χ(x), with energy E. By integrating this equation across x = 0, show that the gradient of the wavefunction jumps across x = 0 according to lim ɛ 0 ( dχ dχ ) (ɛ) dx dx ( ɛ) [You may assume that χ is continuous across x = 0.] = 1 χ(0). (b) Show that there exists a negative energy solution and calculate its energy. (c) Consider a double delta function potential 2 V (x) = [δ(x + a) + δ(x a)]. 2M For sufficiently small, this potential yields a negative energy solution of odd parity, i.e. χ( x) = χ(x). Show that its energy is given by E = 2 2M λ2, where tanh λa = λ 1 λ. [You may again assume χ is continuous across x = ±a.] Part IB
45 /I/9D From the expressions L x = yp z zp y, L y = zp x xp z, L z = xp y yp x, show that (x + iy)z is an eigenfunction of L 2 and L z, and compute the corresponding eigenvalues. Part IB
46 /II/18D Consider a quantum mechanical particle moving in an upsidedown harmonic oscillator potential. Its wavefunction Ψ(x, t) evolves according to the timedependent Schrödinger equation, i Ψ t = 2 Ψ 2 2 x x2 Ψ. (1) (a) Verify that is a solution of equation (1), provided that Ψ(x, t) = A(t)e B(t)x2 (2) da dt = i AB, and db dt = i 2 2i B2. (3) (b) Verify that B = 1 2 tan(φ it) provides a solution to (3), where φ is an arbitrary real constant. (c) The expectation value of an operator O at time t is O (t) dxψ (x, t)oψ(x, t), where Ψ(x, t) is the normalised wave function. Show that for Ψ(x, t) given by (2), Hence show that as t, x 2 = 1 4Re(B), p2 = 4 2 B 2 x 2. x 2 p 2 4 sin 2φ e2t. [Hint: You may use dx e Cx2 x 2 dx e Cx2 = 1 2C.] Part IB
47 /II/20D A quantum mechanical system has two states χ 0 and χ 1, which are normalised energy eigenstates of a Hamiltonian H 3, with H 3 χ 0 = χ 0, H 3 χ 1 = +χ 1. A general timedependent state may be written Ψ(t) = a 0 (t)χ 0 + a 1 (t)χ 1, (1) where a 0 (t) and a 1 (t) are complex numbers obeying a 0 (t) 2 + a 1 (t) 2 = 1. (a) Write down the timedependent Schrödinger equation for Ψ(t), and show that if the Hamiltonian is H 3, then i da 0 dt = a 0, i da 1 dt = +a 1. For the general state given in equation (1) above, write down the probability to observe the system, at time t, in a state αχ 0 + βχ 1, properly normalised so that α 2 + β 2 = 1. (b) Now consider starting the system in the state χ 0 at time t = 0, and evolving it with a different Hamiltonian H 1, which acts on the states χ 0 and χ 1 as follows: H 1 χ 0 = χ 1, H 1 χ 1 = χ 0. By solving the timedependent Schrödinger equation for the Hamiltonian H 1, find a 0 (t) and a 1 (t) in this case. Hence determine the shortest time T > 0 such that Ψ(T ) is an eigenstate of H 3 with eigenvalue +1. (c) Now consider taking the state Ψ(T ) from part (b), and evolving it for further length of time T, with Hamiltonian H 2, which acts on the states χ 0 and χ 1 as follows: H 2 χ 0 = iχ 1, H 2 χ 1 = iχ 0. What is the final state of the system? Is this state observationally distinguishable from the original state χ 0? Part IB
PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 13, 2004
PHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Exam Solutions Dec. 1, 2004 No materials allowed. If you can t remember a formula, ask and I might help. If you can t do one part of a problem,
More informationHermitian Operators An important property of operators is suggested by considering the Hamiltonian for the particle in a box: d 2 dx 2 (1)
CHAPTER 4 PRINCIPLES OF QUANTUM MECHANICS In this Chapter we will continue to develop the mathematical formalism of quantum mechanics, using heuristic arguments as necessary. This will lead to a system
More informationFor the case of an Ndimensional spinor the vector α is associated to the onedimensional . N
1 CHAPTER 1 Review of basic Quantum Mechanics concepts Introduction. Hermitian operators. Physical meaning of the eigenvectors and eigenvalues of Hermitian operators. Representations and their use. onhermitian
More informationRutgers  Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006
Rutgers  Physics Graduate Qualifying Exam Quantum Mechanics: September 1, 2006 QA J is an angular momentum vector with components J x, J y, J z. A quantum mechanical state is an eigenfunction of J 2 J
More informationSolved Problems on Quantum Mechanics in One Dimension
Solved Problems on Quantum Mechanics in One Dimension Charles Asman, Adam Monahan and Malcolm McMillan Department of Physics and Astronomy University of British Columbia, Vancouver, British Columbia, Canada
More informationHarmonic Oscillator Physics
Physics 34 Lecture 9 Harmonic Oscillator Physics Lecture 9 Physics 34 Quantum Mechanics I Friday, February th, 00 For the harmonic oscillator potential in the timeindependent Schrödinger equation: d ψx
More informationThe Essentials of Quantum Mechanics
The Essentials of Quantum Mechanics Prof. Mark Alford v7, 2008Oct22 In classical mechanics, a particle has an exact, sharply defined position and an exact, sharply defined momentum at all times. Quantum
More informationModule 1: Quantum Mechanics  2
Quantum Mechanics  Assignment Question: Module 1 Quantum Mechanics Module 1: Quantum Mechanics  01. (a) What do you mean by wave function? Explain its physical interpretation. Write the normalization
More information1D 3D 1D 3D. is called eigenstate or state function. When an operator act on a state, it can be written as
Chapter 3 (Lecture 45) Postulates of Quantum Mechanics Now we turn to an application of the preceding material, and move into the foundations of quantum mechanics. Quantum mechanics is based on a series
More informationMixed states and pure states
Mixed states and pure states (Dated: April 9, 2009) These are brief notes on the abstract formalism of quantum mechanics. They will introduce the concepts of pure and mixed quantum states. Some statements
More informationWrite your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used.
UNIVERSITY OF LONDON BSc/MSci EXAMINATION June 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship
More information1 The Fourier Transform
Physics 326: Quantum Mechanics I Prof. Michael S. Vogeley The Fourier Transform and Free Particle Wave Functions 1 The Fourier Transform 1.1 Fourier transform of a periodic function A function f(x) that
More information4. The Infinite Square Well
4. The Infinite Square Well Copyright c 215 216, Daniel V. Schroeder In the previous lesson I emphasized the free particle, for which V (x) =, because its energy eigenfunctions are so simple: they re the
More informationChemistry 431. NC State University. Lecture 3. The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics
Chemistry 431 Lecture 3 The Schrödinger Equation The Particle in a Box (part 1) Orthogonality Postulates of Quantum Mechanics NC State University Derivation of the Schrödinger Equation The Schrödinger
More informationUNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet
UNIVERSITETET I OSLO Det matematisknaturvitenskapelige fakultet Solution for take home exam: FYS311, Oct. 7, 11. 1.1 The Hamiltonian of a charged particle in a weak magnetic field is Ĥ = P /m q mc P A
More informationA. The wavefunction itself Ψ is represented as a socalled 'ket' Ψ>.
Quantum Mechanical Operators and Commutation C I. BraKet Notation It is conventional to represent integrals that occur in quantum mechanics in a notation that is independent of the number of coordinates
More information221A Lecture Notes Variational Method
1 Introduction 1A Lecture Notes Variational Method Most of the problems in physics cannot be solved exactly, and hence need to be dealt with approximately. There are two common methods used in quantum
More information3. Some onedimensional potentials
TFY4215/FY1006 Tillegg 3 1 TILLEGG 3 3. Some onedimensional potentials This Tillegg is a supplement to sections 3.1, 3.3 and 3.5 in Hemmer s book. Sections marked with *** are not part of the introductory
More informationFrom Fourier Series to Fourier Integral
From Fourier Series to Fourier Integral Fourier series for periodic functions Consider the space of doubly differentiable functions of one variable x defined within the interval x [ L/2, L/2]. In this
More information1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = (2ψ 1,0,0 3ψ 2,0,0 ψ 3,2,2 )
CHEM 352: Examples for chapter 2. 1. The quantum mechanical state of a hydrogen atom is described by the following superposition: ψ = 1 14 2ψ 1,, 3ψ 2,, ψ 3,2,2 ) where ψ n,l,m are eigenfunctions of the
More informationWe consider a hydrogen atom in the ground state in the uniform electric field
Lecture 13 Page 1 Lectures 1314 Hydrogen atom in electric field. Quadratic Stark effect. Atomic polarizability. Emission and Absorption of Electromagnetic Radiation by Atoms Transition probabilities and
More informationQualification Exam: Quantum Mechanics
Qualification Exam: Quantum Mechanics Name:, QEID#26080663: March, 2014 Qualification Exam QEID#26080663 2 1 Undergraduate level Problem 1. 1983FallQMU1 ID:QMU2 Consider two spin 1/2 particles interacting
More informationPHY4604 Introduction to Quantum Mechanics Fall 2004 Practice Test 3 November 22, 2004
PHY464 Introduction to Quantum Mechanics Fall 4 Practice Test 3 November, 4 These problems are similar but not identical to the actual test. One or two parts will actually show up.. Short answer. (a) Recall
More informationTime dependence in quantum mechanics Notes on Quantum Mechanics
Time dependence in quantum mechanics Notes on Quantum Mechanics http://quantum.bu.edu/notes/quantummechanics/timedependence.pdf Last updated Thursday, November 20, 2003 13:22:3705:00 Copyright 2003 Dan
More informationMITES 2010: Physics III Survey of Modern Physics Final Exam Solutions
MITES 2010: Physics III Survey of Modern Physics Final Exam Solutions Exercises 1. Problem 1. Consider a particle with mass m that moves in onedimension. Its position at time t is x(t. As a function of
More informationThe Quantum Harmonic Oscillator Stephen Webb
The Quantum Harmonic Oscillator Stephen Webb The Importance of the Harmonic Oscillator The quantum harmonic oscillator holds a unique importance in quantum mechanics, as it is both one of the few problems
More informationQuick Reference Guide to Linear Algebra in Quantum Mechanics
Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................
More information1. Thegroundstatewavefunctionforahydrogenatomisψ 0 (r) =
CHEM 5: Examples for chapter 1. 1. Thegroundstatewavefunctionforahydrogenatomisψ (r = 1 e r/a. πa (a What is the probability for finding the electron within radius of a from the nucleus? (b Two excited
More informationFourier Series and SturmLiouville Eigenvalue Problems
Fourier Series and SturmLiouville Eigenvalue Problems 2009 Outline Functions Fourier Series Representation Halfrange Expansion Convergence of Fourier Series Parseval s Theorem and Mean Square Error Complex
More informationCHAPTER 5 THE HARMONIC OSCILLATOR
CHAPTER 5 THE HARMONIC OSCILLATOR The harmonic oscillator is a model which has several important applications in both classical and quantum mechanics. It serves as a prototype in the mathematical treatment
More informationSymbols, conversions, and atomic units
Appendix C Symbols, conversions, and atomic units This appendix provides a tabulation of all symbols used throughout this thesis, as well as conversion units that may useful for reference while comparing
More informationBasic Quantum Mechanics
Basic Quantum Mechanics Postulates of QM  The state of a system with n position variables q, q, qn is specified by a state (or wave) function Ψ(q, q, qn)  To every observable (physical magnitude) there
More informationCHAPTER 6 THE HYDROGEN ATOM OUTLINE. 3. The HydrogenAtom Wavefunctions (Complex and Real)
CHAPTER 6 THE HYDROGEN ATOM OUTLINE Homework Questions Attached SECT TOPIC 1. The Hydrogen Atom Schrödinger Equation. The Radial Equation (Wavefunctions and Energies) 3. The HydrogenAtom Wavefunctions
More information2 Creation and Annihilation Operators
Physics 195 Course Notes Second Quantization 030304 F. Porter 1 Introduction This note is an introduction to the topic of second quantization, and hence to quantum field theory. In the Electromagnetic
More informationAppendix E  Elements of Quantum Mechanics
1 Appendix E  Elements of Quantum Mechanics Quantum mechanics provides a correct description of phenomena on the atomic or sub atomic scale, where the ideas of classical mechanics are not generally applicable.
More informationLecture VI. Review of even and odd functions Definition 1 A function f(x) is called an even function if. f( x) = f(x)
ecture VI Abstract Before learning to solve partial differential equations, it is necessary to know how to approximate arbitrary functions by infinite series, using special families of functions This process
More informationThe Schrödinger Equation
The Schrödinger Equation When we talked about the axioms of quantum mechanics, we gave a reduced list. We did not talk about how to determine the eigenfunctions for a given situation, or the time development
More informationMotion of a pointlike particle with a single degree of freedom
Chapter 3 Motion of a pointlike particle with a single degree of freedom 3.1 Continuous observables In this chapter, we study basic quantum physics of the simplest mechanical system: translational motion
More informationBead moving along a thin, rigid, wire.
Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing
More informationConceptual Approaches to the Principles of Least Action
Conceptual Approaches to the Principles of Least Action Karlstad University Analytical Mechanics FYGB08 January 3, 015 Author: Isabella Danielsson Supervisors: Jürgen Fuchs Igor Buchberger Abstract We
More informationLecture 18: Quantum Mechanics. Reading: Zumdahl 12.5, 12.6 Outline. Problems (Chapter 12 Zumdahl 5 th Ed.)
Lecture 18: Quantum Mechanics Reading: Zumdahl 1.5, 1.6 Outline Basic concepts of quantum mechanics and molecular structure A model system: particle in a box. Demos how Q.M. actually obtains a wave function.
More informationPART A: THREE MARK QUESTIONS
PART A: THREE MARK QUESTIONS. If J x, J y, J z are angular momentum operators, the eigenvalues of the operator J x + J y )/ h are real and discrete with rational spacing real and discrete with irrational
More informationLecture 18 Timedependent perturbation theory
Lecture 18 Timedependent perturbation theory Timedependent perturbation theory So far, we have focused on quantum mechanics of systems described by Hamiltonians that are timeindependent. In such cases,
More information develop a theory that describes the wave properties of particles correctly
Quantum Mechanics Bohr's model: BUT: In 192526: by 1930s:  one of the first ones to use idea of matter waves to solve a problem  gives good explanation of spectrum of single electron atoms, like hydrogen
More informationFourier Series. A Fourier series is an infinite series of the form. a + b n cos(nωx) +
Fourier Series A Fourier series is an infinite series of the form a b n cos(nωx) c n sin(nωx). Virtually any periodic function that arises in applications can be represented as the sum of a Fourier series.
More informationFrom Einstein to KleinGordon Quantum Mechanics and Relativity
From Einstein to KleinGordon Quantum Mechanics and Relativity Aline Ribeiro Department of Mathematics University of Toronto March 24, 2002 Abstract We study the development from Einstein s relativistic
More informationFermi s Golden Rule. Emmanuel N. Koukaras A.M.: 198
Fermi s Golden Rule Emmanuel N. Kouaras A.M.: 198 Abstract We present a proof of Fermi s Golden rule from an educational perspective without compromising formalism. 2 1. Introduction Fermi s Golden Rule
More information5. Spherically symmetric potentials
TFY4215/FY1006 Tillegg 5 1 TILLEGG 5 5. Spherically symmetric potentials Chapter 5 of FY1006/TFY4215 Spherically symmetric potentials is covered by sections 5.1 and 5.4 5.7 in Hemmer s bok, together with
More information2. Wavefunctions. An example wavefunction
2. Wavefunctions Copyright c 2015 2016, Daniel V. Schroeder To create a precise theory of the wave properties of particles and of measurement probabilities, we introduce the concept of a wavefunction:
More informationHomework One Solutions. Keith Fratus
Homework One Solutions Keith Fratus June 8, 011 1 Problem One 1.1 Part a In this problem, we ll assume the fact that the sum of two complex numbers is another complex number, and also that the product
More information1 Lecture 3: Operators in Quantum Mechanics
1 Lecture 3: Operators in Quantum Mechanics 1.1 Basic notions of operator algebra. In the previous lectures we have met operators: ˆx and ˆp = i h they are called fundamental operators. Many operators
More informationNotes on wavefunctions III: time dependence and the Schrödinger equation
Notes on wavefunctions III: time dependence and the Schrödinger equation We now understand that the wavefunctions for traveling particles with momentum p look like wavepackets with wavelength λ = h/p,
More informationQ ( q(m, t 0 ) n) S t.
THE HEAT EQUATION The main equations that we will be dealing with are the heat equation, the wave equation, and the potential equation. We use simple physical principles to show how these equations are
More informationIntroduction to Green s Functions: Lecture notes 1
October 18, 26 Introduction to Green s Functions: Lecture notes 1 Edwin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE16 91 Stockholm, Sweden Abstract In the present notes I try to give a better
More informationLectureXXIV. Quantum Mechanics Expectation values and uncertainty
LectureXXIV Quantum Mechanics Expectation values and uncertainty Expectation values We are looking for expectation values of position and momentum knowing the state of the particle, i,e., the wave function
More informationPhysics 53. Wave Motion 1
Physics 53 Wave Motion 1 It's just a job. Grass grows, waves pound the sand, I beat people up. Muhammad Ali Overview To transport energy, momentum or angular momentum from one place to another, one can
More informationCHEM344 HW#7 Due: Fri, Mar BEFORE CLASS!
CHEM344 HW#7 Due: Fri, Mar 14@2pm BEFORE CLASS! HW to be handed in: Atkins Chapter 8: Exercises: 8.11(b), 8.16(b), 8.19(b), Problems: 8.2, 8.4, 8.12, 8.34, Chapter 9: Exercises: 9.5(b), 9.7(b), Extra (do
More informationElectromagnetic Waves
May 4, 2010 1 1 J.D.Jackson, Classical Electrodynamics, 2nd Edition, Section 7 Maxwell Equations A basic feature of Maxwell equations for the EM field is the existence of travelling wave solutions which
More informationEUF. Joint Entrance Examination for Postgraduate Courses in Physics
EUF Joint Entrance Examination for Postgraduate Courses in Physics For the second semester 204 23 April 204 Part Instructions Do not write your name on the test. It should be identified only by your candidate
More informationQuantum Mechanics. Professor N. Dorey. Copyright 2008 University of Cambridge. Not to be quoted or reproduced without permission.
Mathematical Tripos IB Michaelmas 2007 Quantum Mechanics Professor N. Dorey DAMTP, University of Cambridge email: n.dorey@damtp.cam.ac.uk Recommended books S. Gasiorowicz, Quantum Physics, Wiley 2003.
More informationThe force equation of quantum mechanics.
The force equation of quantum mechanics. by M. W. Evans, Civil List and Guild of Graduates, University of Wales, (www.webarchive.org.uk, www.aias.us,, www.atomicprecision.com, www.upitec.org, www.et3m.net)
More informationRelativistic Electromagnetism
Chapter 8 Relativistic Electromagnetism In which it is shown that electricity and magnetism can no more be separated than space and time. 8.1 Magnetism from Electricity Our starting point is the electric
More informationUNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH College of Informatics and Electronics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4607 SEMESTER: Autumn 200405 MODULE TITLE: Fluid mechanics DURATION OF EXAMINATION:
More informationFourier Series. Chapter Some Properties of Functions Goal Preliminary Remarks
Chapter 3 Fourier Series 3.1 Some Properties of Functions 3.1.1 Goal We review some results about functions which play an important role in the development of the theory of Fourier series. These results
More informationψ 2 ψ 3 h 2 d 2 2m dx 0 x > a V (x) = V0 x a x < a = B sin kx or B'cos kx a x a = Ce κx x > a k =, κ = h Since the solution has to be odd, ψ 2
.0 Proble Set 3 Solution.Solution The energy eigenvalue proble for the given syste is: Ĥψ = Eψ h d Ĥ = + V (x) dx 0 x > a V (x) = V0 x a For bounded particle, the solution is: ψ = Ae κx x < a ψ = B sin
More informationChemistry 417! 1! Fall Chapter 2 Notes
Chemistry 417! 1! Fall 2012 Chapter 2 Notes September 3, 2012! Chapter 2, up to shielding 1. Atomic Structure in broad terms a. nucleus and electron cloud b. nomenclature, so we may communicate c. Carbon12
More informationQuantum Mechanics. December 17, 2007
Quantum Mechanics Based on lectures given by J.Billowes at the University of Manchester SeptDec 7 Please email me with any comments/corrections: jap@watering.co.uk J.Pearson December 7, 7 Contents Review.
More informationFermi s golden rule. 1 Main results
Fermi s golden rule Andreas Wacker 1 Mathematical Physics, Lund University October 1, 216 Fermi s golden rule 2 is a simple expression for the transition probabilities between states of a quantum system,
More informationbe a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that
Problem 1A. Let... X 2 X 1 be a nested sequence of closed nonempty connected subsets of a compact metric space X. Prove that i=1 X i is nonempty and connected. Since X i is closed in X, it is compact.
More informationLecture 2: Angular momentum and rotation
Lecture : Angular momentum and rotation Angular momentum of a composite system Let J and J be two angular momentum operators. One might imagine them to be: the orbital angular momentum and spin of a particle;
More informationMATH 461: Fourier Series and Boundary Value Problems
MATH 461: Fourier Series and Boundary Value Problems Chapter III: Fourier Series Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2015 fasshauer@iit.edu MATH 461 Chapter
More informationHigh Energy Physics Summer School 2016 PreSchool Notes and Problems
STFC HEP Summer School September 2016 High Energy Physics Summer School 2016 PreSchool Notes and Problems Quantum Field Theory Christoph Englert, University of Glasgow Introduction to QED and QCD Andrea
More informationProblems in Quantum Mechanics
Problems in Quantum Mechanics Daniel F. Styer July 1994 Contents 1 SternGerlach Analyzers 1 Photon Polarization 5 3 Matrix Mathematics 9 4 The Density Matrix 1 5 Neutral K Mesons 13 6 Continuum Systems
More informationConcepts for specific heat
Concepts for specific heat Andreas Wacker, Matematisk Fysik, Lunds Universitet Andreas.Wacker@fysik.lu.se November 8, 1 1 Introduction In this notes I want to briefly eplain general results for the internal
More informationPhysics 70007, Fall 2009 Solutions to HW #4
Physics 77, Fall 9 Solutions to HW #4 November 9. Sakurai. Consider a particle subject to a onedimensional simple harmonic oscillator potential. Suppose at t the state vector is given by ipa exp where
More informationNumerical Simulations of HighDimensional ModeCoupling Models in Molecular Dynamics
Dickinson College Dickinson Scholar Honors Theses By Year Honors Theses 5222016 Numerical Simulations of HighDimensional ModeCoupling Models in Molecular Dynamics Kyle Lewis Liss Dickinson College
More informationThe Schrödinger Equation. Erwin Schrödinger Nobel Prize in Physics 1933
The Schrödinger Equation Erwin Schrödinger 18871961 Nobel Prize in Physics 1933 The Schrödinger Wave Equation The Schrödinger wave equation in its timedependent form for a particle of energy E moving
More informationCHAPTER 12 MOLECULAR SYMMETRY
CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical
More informationINTRODUCTION TO FOURIER TRANSFORMS FOR PHYSICISTS
INTRODUCTION TO FOURIER TRANSFORMS FOR PHYSICISTS JAMES G. O BRIEN As we will see in the next section, the Fourier transform is developed from the Fourier integral, so it shares many properties of the
More informationMathematical Formulation of the Superposition Principle
Mathematical Formulation of the Superposition Principle Superposition add states together, get new states. Math quantity associated with states must also have this property. Vectors have this property.
More informationQuantum Mechanics. DungHai Lee
Quantum Mechanics DungHai Lee Summer 2000 Contents 1 A brief reminder of linear Algebra 3 1.1 Linear vector space..................... 3 1.2 Linear operators and their corresponding matrices.... 5 1.3
More informationQuantum Mechanics: Postulates
Quantum Mechanics: Postulates 5th April 2010 I. Physical meaning of the Wavefunction Postulate 1: The wavefunction attempts to describe a quantum mechanical entity (photon, electron, xray, etc.) through
More informationProblem 1 (10 pts) Find the radius of convergence and interval of convergence of the series
1 Problem 1 (10 pts) Find the radius of convergence and interval of convergence of the series a n n=1 n(x + 2) n 5 n 1. n(x + 2)n Solution: Do the ratio test for the absolute convergence. Let a n =. Then,
More informationclassical vs. quantum statistics, quasiprobability distributions
Lecture 6: Quantum states in phase space classical vs. quantum statistics, quasiprobability distributions operator expansion in phase space Classical vs. quantum statistics, quasiprobability distributions:
More information3. A LITTLE ABOUT GROUP THEORY
3. A LITTLE ABOUT GROUP THEORY 3.1 Preliminaries It is an apparent fact that nature exhibits many symmetries, both exact and approximate. A symmetry is an invariance property of a system under a set of
More informationChapter 9 Unitary Groups and SU(N)
Chapter 9 Unitary Groups and SU(N) The irreducible representations of SO(3) are appropriate for describing the degeneracies of states of quantum mechanical systems which have rotational symmetry in three
More informationPhysics 505 Fall 2007 Homework Assignment #2 Solutions. Textbook problems: Ch. 2: 2.2, 2.8, 2.10, 2.11
Physics 55 Fall 27 Homework Assignment #2 Solutions Textbook problems: Ch. 2: 2.2, 2.8, 2., 2. 2.2 Using the method of images, discuss the problem of a point charge q inside a hollow, grounded, conducting
More informationMatter Waves. Solutions of Selected Problems
Chapter 5 Matter Waves. Solutions of Selected Problems 5. Problem 5. (In the text book) For an electron to be confined to a nucleus, its de Broglie wavelength would have to be less than 0 4 m. (a) What
More informationTangent and normal lines to conics
4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints
More informationNear horizon black holes in diverse dimensions and integrable models
Near horizon black holes in diverse dimensions and integrable models Anton Galajinsky Tomsk Polytechnic University LPI, 2012 A. Galajinsky (TPU) Near horizon black holes and integrable models LPI, 2012
More informationQUANTUM MECHANICS (PHYS4010) LECTURE NOTES
QUANTUM MECHANICS PHYS4 LECTURE NOTES Lecture notes based on a course given by Roman Koniuk. The course begins with a formal introduction into quantum mechanics and then moves to solving different quantum
More informationChem 81 Fall, Exam 2 Solutions
Chem 81 Fall, 1 Exam Solutions 1. (15 points) Consider the boron atom. (a) What is the ground state electron configuration for B? (If you don t remember where B is in the periodic table, ask me. I ll tell
More informationThomson and Rayleigh Scattering
Thomson and Rayleigh Scattering In this and the next several lectures, we re going to explore in more detail some specific radiative processes. The simplest, and the first we ll do, involves scattering.
More informationGround State of the He Atom 1s State
Ground State of the He Atom s State First order perturbation theory Neglecting nuclear motion H m m 4 r 4 r r Ze Ze e o o o o 4 o kinetic energies attraction of electrons to nucleus electron electron repulsion
More informationFLAP P11.2 The quantum harmonic oscillator
F L E X I B L E L E A R N I N G A P P R O A C H T O P H Y S I C S Module P. Opening items. Module introduction. Fast track questions.3 Ready to study? The harmonic oscillator. Classical description of
More informationPHY411. PROBLEM SET 3
PHY411. PROBLEM SET 3 1. Conserved Quantities; the RungeLenz Vector The Hamiltonian for the Kepler system is H(r, p) = p2 2 GM r where p is momentum, L is angular momentum per unit mass, and r is the
More information1 Variational calculation of a 1D bound state
TEORETISK FYSIK, KTH TENTAMEN I KVANTMEKANIK FÖRDJUPNINGSKURS EXAMINATION IN ADVANCED QUANTUM MECHAN ICS Kvantmekanik fördjupningskurs SI38 för F4 Thursday December, 7, 8. 13. Write on each page: Name,
More informationLévy path integral approach to the fractional Schrödinger equation with deltaperturbed infinite square well
7 th Jagna International Workshop (204 International Journal of Modern Physics: Conference Series Vol. 36 (205 56005 (5 pages c The Authors DOI: 0.42/S200945560050 Lévy path integral approach to the fractional
More informationThomson and Rayleigh Scattering
Thomson and Rayleigh Scattering Initial questions: What produces the shapes of emission and absorption lines? What information can we get from them regarding the environment or other conditions? In this
More informationPhysics 53. Wave Motion 2. If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it.
Physics 53 Wave Motion 2 If at first you don't succeed, try, try again. Then quit. No use being a damn fool about it. W.C. Fields Waves in two or three dimensions Our description so far has been confined
More information